help-36

[rancid zenith]

Not to mention that it is redundant

By definition

largest factor smaller than root n < root n < smallest factor bigger than root n

So you didnt need to include that

[raw merlin]

lol yeah

[rancid zenith]

Anyways it seems like you got the idea but this proof is quite unreadable for me

[raw merlin]

what can you not read or understand

<@&286206848099549185>

[fast smelt]

So, to clarify. Let $n \in \bN$ be a natural number. We define $f^-_n(k)$ as the number of factors of $n$ less than $k$, and $f^+_n(k)$ as the number of factors of $n$ greater than $k$. Let $k \in \bR$ be a real number such that $f^-_n(k) = f^+_n(k)$. Let $p \in \bN$ be a factor of $n$ such that $p < k$. Prove $n > pk$.

[soft zealot]

Shuba

[raw merlin]

yes

[fast smelt]

If p is a factor of n, then n/p is also a factor of n. What can we say about n/p?

remember, k is right in the middle of all the factors of n.

[raw merlin]

n/p >= root(n)

[fast smelt]

Yes.
Can you prove it?
What does that mean for k?

[raw merlin]

p < root(n)
1/root(n) < 1/p
root(n) < n/p

Well the greatest factor lesser than sqrt(n) < k < smallest factor bigger than sqrt(n)

[fast smelt]

yep, you're an the right track

[raw merlin]

the smallest factor bigger than sqrt(n) <= n/p

[fast smelt]

Here's what you need to do.
\begin{enumerate}
\item Let $k = \sqrt{n}$.
\item Prove $f^-_n(k) = f^+_n(k)$.
\item Prove $$\forall p \in \bN .; p | n \implies p < k \implies n/p > k \implies n > pk$$
\end{enumerate}

[soft zealot]

Shuba

[fast smelt]

Let us know if you want more help with either of these steps 🙂

[raw merlin]

What does => mean

[fast smelt]

implies

Point 3, in english, is "for all p, p is a factor of n where p < k implies n/p > k implies `n > pk".

[raw merlin]

let k = root(n) ?

[fast smelt]

I've bungled the maths in that TeXit a bit. Sorry. I think you understand what you need to do though

[fast smelt]

Yep. We want to find a k such that the number of factors of n less than k equals the number of factors of n greater than k. A good candidate for this is k = root(n). Now, we need to prove the indeed meets the criteria for k.

Then, once we've verified k, we can show if any factor p < k, then n/p > k

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[tulip atlas]

<@&286206848099549185>

[sage meteor]

!15min

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[gusty ravine]

wherever delta x is zero for that time v is also zero

[sage meteor]

you're pinged already so grrr the average velocity from 0 to t where it is zero is the point where the secant line is parallel to x axis

i think

[gusty ravine]

in this case tfinal is where you find the delta x as zwerro or two same x values

[gusty ravine]

but that is instanttaneous

[lyric obsidian]

notice that if the average velocity is zero, then what must the xposition be

[lyric obsidian]

!nosols

[gusty ravine]

im talkinbout the av

[final saddle]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[gusty ravine]

okae

[sage meteor]

no instantanenous is the tangent line

for this instance

starting from 0 to a random point on the curve

find the secand

parallel to x axis

(which is again, close to 10 seconds)

[lyric obsidian]

vro

[gusty ravine]

no no im just v(av) = delta x/deltaT

[lyric obsidian]

could you at least spoil that??

[gusty ravine]

BUT THAT IS INSTANTANEOUS

[gusty ravine]

HOWW

[sage meteor]

ong

i'm just gonna

grrr

[gusty ravine]

naww

the tangent at any point to the curve represents the dx/dt or inst v

[lyric obsidian]

It isn't.

[gusty ravine]

sorry

i forgor the secand

[lyric obsidian]

@sage meteor Please put your suspected answer in spoilers.

[tiny barn]

What is the secant line can u explain I’ve never heard of this before (or I didn’t pay attention in my calc lectures)

[lyric obsidian]

!occupied

[final saddle]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[gusty ravine]

yeaa exactly wherever ||the secand is parallel to x axis but @sage meteor that would mean the exact same thing as what i said where delta x = zero since they both lie along the same line which has slope zero||

[gusty ravine]

how to

please tell

[lyric obsidian]

Just use ||message||

||message||

[sage meteor]

okay whatever, same same

[sage meteor]

line joining two points on a curve

that's all

[lyric obsidian]

Yes

[sage meteor]

like in teh diagram

[tiny barn]

Thx

[final saddle]

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@tulip atlas Has your question been resolved?

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[distant scarab]

,, A_n^k \cdot A_b^c = A_{n+b}^{k+c}

[soft zealot]

<rajel />

[distant scarab]

is this always correct

A in probabilites , forgot what it was called

but it goes $A_n^k = \frac{n!}{(n-k)!}$

[soft zealot]

<rajel />

[tired walrus]

this is usually called nPk in english

for permutation

anyway this looks sus though

i can tell you this is wrong for at least one set of values

[distant scarab]

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[last cape]

1001000100001.....
Is this an irrational number

[onyx peak]

did you mean 0.1001000100001.... ?

if so, then yes

all rational numbers have (eventually) periodic expansion

this doesnt have eventually periodic expansion

therefore it's not rational

[last cape]

No, 1001000100001.....

[bold turtle]

(that or its decimal expansion terminates)

[desert mantle]

thats not any number

[desert mantle]

(aka eventually periodic 0)

[bold turtle]

You're asking about a whole number with infinitely many digits?

[last cape]

yea

[dapper hull]

I wouldn’t say that’s enough info to decide

[left echo]

anti-2adic

[last cape]

thereis a pattern

[bold turtle]

I see we've finally found the maths equivalent of "anti spiral"

But assuming this is just a whole number, then... well it's a whole number so it's rational

It's a functionally useless number quand-même, but still

[dapper hull]

A pattern isn’t necessarily enough

[last cape]

oh

good

[last cape]

why

[dapper hull]

One example

A pattern is there but still it’s not rational

[last cape]

bruh

[tired walrus]

hold on though like what is this even supposed to mean

you're gonna have to explain to us peons what your "integer but the digits go infinitely to the left" notation means

[last cape]

how a pattern implies rationality

[desert mantle]

a periodic pattern implies rationality

but not this

and again, this isnt even a number

[left echo]

"pattern" doesn't really have a mathematical meaning in this context

we are specifically looking for blocks of repeating digits

[desert mantle]

not in any usual sense of the word

not even a p-adic number

its just nonsense

[tired walrus]

which, again, brings us to the main point: what do YOU mean with this notation?

[eager shore]

not like theres an interpretation with a finite answer

[last cape]

It is an infinite sum like 1 + 2 + 3 + 4 + .....

= -1/12

which is rational

[desert mantle]

thats just not how any of that works

[tired walrus]

... well ok then can you spell out your infinite sum for us in more conventional, non-genius-friendly notation?

also yeah no. cut it with the -1/12 BS.

[tired walrus]

until you do this, we'll just have to assume this is either a gloat or an ego-stroke.

[bleak granite]

you're saying zero point 1 followed by 2 zeroes, then a 1 followed by 3 zeroes, etc. (generally a 1 followed by n zeroes)?

[desert mantle]

read the channel. no he does not mean that

[bleak granite]

ah did not load fully for me, sorry :)

[onyx peak]

Idk if it's a joke or not, but 1 + 2 + 3 + 4 + ... diverges. It's certainly not -1/12

[tired walrus]

shhhhh he's clearly ramanujan II /s

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[last cape]

Thanks.

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[vague token]

does anyone know any google lassroom codes for stem groups

[final saddle]

Please don't occupy multiple help channels.

[vital crag]

Uhhh this isn't what help channels are for

[vague token]

oh

..........

sorry abt that then

[steel talon]

!done

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[vague token]

.reopen

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✅

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[lost sphinx]

Is this right?

[lost sphinx]

Actually wait

B

I think it’s B

Is B right 😅

[bleak granite]

this seems like an economics question

[tiny gorge]

flour gets more expensive and bread gets cheaper as a result?

[lost sphinx]

It is! 🙂

Wait yeah what

Hold up

So it’s A

I’m loosing it sorry on 3 hours of sleep

[dense coral]

if you have cheaper flour, you get more bread and cheaper bread

[bleak granite]

the movement of equilibrium point described by the question is a result of shift left of demand

[lost sphinx]

Yes

Thank you

[dense coral]

I don't actually know any economics, but this is the only reasonable answer to me

[lost sphinx]

Thank you guys so much

[dense coral]

an increase in the price of rye bread shouldn't make white bread cheaper, I feel?

[lost sphinx]

Yes I did A and it was right

[dense coral]

there'd be more demand for it, but like

[lost sphinx]

Yes and it would be a demand thing

Yup

[dense coral]

[lost sphinx]

I hate Econ

Ty yall

[tiny gorge]

more demand but not more supply, so the price of white bread should go up too

[dense coral]

mhm

[tiny gorge]

(maybe not as much as rye)

[lost sphinx]

Yes

Econ can be so much of everything so I get triggered

.close

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[storm haven]

the underlined ones 😭 except for the first one, i alr get that

[storm haven]

just the next perfect square + why c =/= 1,2,4,5

[final saddle]

@storm haven Has your question been resolved?

[storm haven]

<@&286206848099549185>

[jagged flare]

c=/=2 and c=/=5 is cause its always increasing (there is no square residue 2 mod 3)

for the next perfect square it was the mod 3 reasoning as yesterday

[pale niche]

at least one of $(\sqrt{c}+d)^2)$ must be 0 mod 3 for $d \in {1,2,3}$

[jagged flare]

no way gci reference??

[pale niche]

lol whats gci

[jagged flare]

ok wished too much

[jagged flare]

grass cutting ubcremental, one of the characters is called grassman

[pale niche]

lol my name is just from the cryptids

[storm haven]

I dont really get it tbh

[brittle breach]

What's the matter?

[storm haven]

Here, i dont understand the 2nd underlined one

I understand everything else

[brittle breach]

A sequence is a pattern of numbers

Arithmetic progression is a + (n-1)d

This is about continuity

That's why we take c to be an input on real line

[brittle breach]

Breaking things down like this will help

[storm haven]

didnt it say integers??

[storm haven]

its fine, i understand the basics

justt the underlined things in the first picture

[brittle breach]

Sorry

[storm haven]

yep its fine

[brittle breach]

In the question, the taken variable C, representing function A0 is greater than 1 and since the questions caps at 3, there can't be any number greater than 3 or less than 0

So 0, 1 and 4 is out of question

[storm haven]

sorry for the late reply

i donnt understand how they got the next square after c

the (floor(root(c)) + 1)^2

oh, and what do you mean by "the question caps at 3"

Guys ping me if reply

[brittle breach]

Mod 3 means 0 to 3

[storm haven]

Oh yeah

[brittle breach]

Since A0 is less than 1, we get 2 or 3

[storm haven]

what do you mean by A0?

A or a_0?

[brittle breach]

It's in the question. Probably a variable relating to 0

[storm haven]

huh

a_0 ?

a sub 0 or capital A?

[brittle breach]

a is the variable for arithmetic progression

[storm haven]

yeah

[brittle breach]

0 is subscript yes

[storm haven]

oh a_0

that is the thing we are looking for

[brittle breach]

True

Maybe

[storm haven]

well, how did tthey figure out the next square in the sequence?

?

[brittle breach]

Note

[storm haven]

huh

[brittle breach]

We don't have to define the sequence, it is already defined in the question

[storm haven]

yep

then?

[brittle breach]

So first check for continuity

[storm haven]

i just need to know where this comes from

[storm haven]

what "continuity"?

[brittle breach]

F(c) = Lim x -> c f(x)

[storm haven]

huh

how will that be relevant?

to the next square

plus doesnt contunuity only work for reals? we are working with integers here

[brittle breach]

You have to check whether they approach from the left hand limit and right hand limit and f(c) is the same .-.

[storm haven]

how is that rellevant?

and we are working with integers

[brittle breach]

Continuity works for even trigonometric functions

I believe

[storm haven]

ofc 💀

cause trig works over the reals

this sequence works over integers

[brittle breach]

Oh

[storm haven]

well, can you just explain where this comes from?

[brittle breach]

It's true that integers are discontinuous

So we don't have to find continuity lol

[storm haven]

yep

[storm haven]

err, @brittle breach could you just explain this?

thats all i need, no continuity or other things

[brittle breach]

Yes

[storm haven]

ok 👍

thank you in advance, and ping me after you type

[brittle breach]

The periodicity of 3 is the repetition of a_n + 3

[brittle breach]

Sure

Lol

Since we add 3 repeatedly, it will raise up to infinity

[storm haven]

hmmm, if it never hits a square right?

as in, a_0 is 2 mod 3?

if it hits a square, it wont blow up

[brittle breach]

It will never hit a square

[storm haven]

huh??

how about 3-> 6-> 9

it hits a square?

or did you mean the a_0 = 2 mod 3 case?

[brittle breach]

Yes

[storm haven]

ah, sorry 😭

well, but thats the easy case

[storm haven]

in the paragraph with the circled things, they assum a_0 =/= 2 mmod 3

[brittle breach]

That means when we don't consider 2, the perfect squares would be the squares of sqroot. C + 1, 2, and 3

[storm haven]

yes, but why

[brittle breach]

Which is a logical way to calculate since we already coined that it can never hit a square

[storm haven]

but it wont hit a square if and only if a_0 is 2 mod 3, but they assumed a_0 is not 2 mod 3?

[storm haven]

wait normie, i dont understand you

[brittle breach]

So if we use it as 3 itself

[storm haven]

waiittt

[brittle breach]

Here's the main catch

[storm haven]

reply to mmy message

[brittle breach]

3 itself will give you the perfect square I guess

[storm haven]

well, reply to this?

[storm haven]

can you elaborate on this? Wont it have to hit a square if a_0 is not 2 mod 3?

[brittle breach]

If it's using 2 in mod 3, it would never hit a square. But likelihood of hitting a square increases if we use 3 in mod 3, as in those places you underlined

[storm haven]

yesh

so, how did they derive the "next sqyare" thing that i circled

i dont get it still..

[brittle breach]

You mean how they get there?

[storm haven]

yeah

how did they derive the circled thing??

[brittle breach]

It's simple, it's a logical method. When using 3 which is the only remaining value, canceling out 0, 1 and 2, we now add constants 1 , 2 and 3 to square root of a_n and find their perfect squares by squaring them

[storm haven]

sorry, i dont get it 😭

why is 3 the only remmaining value?

[brittle breach]

Determine all the values of a_0 for which there is a number X such that a_n = X for infinitely many values of n

That's the real question

[storm haven]

umhm, yes.

and i understand the proof except for 1 thing

i just mneed the "next square" thing

thats it

[brittle breach]

Here's some reasoning

[storm haven]

rememmber to ping me when you reply

i need to soolve other math questions

[brittle breach]

When we find a perfect square there like you reasoned earlier, the sequence is smaller

[storm haven]

ok, i get it

[brittle breach]

When we don't find a perfect square there, then the sequence will become larger to reach the perfect square

[storm haven]

yes

[brittle breach]

Can also lead to infinity

[storm haven]

mmhmmm

[brittle breach]

So there you have it

[storm haven]

??

how do you derive tthe next square with that?

(floor(root(c)) + 1)^2

or 1 can be 2 or 3

[brittle breach]

If an is 0 in mod 3

Then the square root of an must be an integer

[storm haven]

hmm, why?

[brittle breach]

If an is a perfect square

[storm haven]

oh ofc

[brittle breach]

But if an is not a perfect square

Then see this

[storm haven]

👍 ill wait for ur immage / typing

[brittle breach]

a_n+1 = a_n + 3 = 0 (mod 3)

[storm haven]

why?

[brittle breach]

Now consider when an is 0 or 1

[storm haven]

wait, isnt this false?

3 + 1 =/= 3 + 3 mod 3

in the case a_n = 3

[brittle breach]

For A to repeat infinitesimally

We need a cycle

For that, we need to consider that an is NOT a perfect square

[storm haven]

i dont understand

[brittle breach]

Then it would increase by Y, Y + 3, Y + 6, Y + 9

[storm haven]

sorry, @brittle breach its been like 2 hours and weve been getting nowhere

[brittle breach]

Sorry, X

[storm haven]

i think ill just close the help channel

[brittle breach]

.close

[storm haven]

it feels like what your saying is wrong,

[brittle breach]

Not my channel

[storm haven]

.close

[final saddle]

Channel closed

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[sonic crystal]

im confused here it says we set z = to some constant + or - C but earlier it said C>0 or is this only for x^2 + y^2 = C and not x^2 + y^z - z^2??

[terse crypt]

they are different surfaces

[storm haven]

Cause that z not z^2

Cause +- C squared is always positive

[terse crypt]

first one:
[ x^2 + y^2 - z^2 = 1]
second one:
[ x^2 + y^2 = z = f(x,y)]

[soft zealot]

k

[sonic crystal]

oh

true

very true

.close

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[hallow fable]

help

[hallow fable]

so im coding

in a random number generator

its used for a rng game

so set ranint(1;100000)

and given the chances of 20 identity

which the chances goes as 1/2n

and i have to find the between value for each

entity

nvm i got it now

ty yall

.close

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[tough shard]

I have two variables: z_geometry and z_field. z_field is located at z = a, and z_geometry is at z = -a. E is the electric field z component. Is my reasoning right here? I wanna get E(z_geometry) in terms of E(z_field). (also assume E(-z) = -E(z), electric field points upward at z = a and downward at z = -a, z_geometry = -z_field).

[magic coyote]

If this line is correct and ur assumption is correct

Then you're right

Idt i have enough info here to check if your first line is correct

[tough shard]

z_geometry = -z_field

i forgot to add that

wait nvm i added it

what's missing?

the E-field direction?

[magic coyote]

Hm idk wait for a different helper am not sure about this

[cunning marten]

are these supposed to be boundary conditions?

[tough shard]

wdym?

[final saddle]

@tough shard Has your question been resolved?

[magic coyote]

You can ping @Helpers now if you wish

[brittle breach]

If E Z geometry is - E Z field. Then E - Z field is E Z field.
But it looks like if your image held true then at the second step, something must have already been calculated

@tough shard

[tough shard]

Shouldn't there be a negative sign on line 4?

[brittle breach]

How do we know when the second line is incorrect

[tough shard]

since E(z_geometry) point downward, E(z_field) points upward

why is it incorrect

E(z) is an odd function

[brittle breach]

OK so we don't use basic algebra methodologies here?

[tough shard]

E(-z_field) = -E(z_field)

since E(z_field) is odd

[brittle breach]

= E (z field)
Minus cancels minus so there is no more minus behind E

[tough shard]

yea but, at the fourth step, E(z_field) points upward, E(z_geometry) points downward, so we can't have E(z_geometry) = E(z_field), we gotta include a minus sign, don't you think so? Like we included a minus sign at the first step

[final saddle]

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[raw escarp]

How can i simplify this / why is the answer 1

[zenith cedar]

Hmmm a method we can use is take it as x

And cube it

So x³ is this expression cubed

[spiral marsh]

i'd say

take those two as a and b

and use the formula for (a+b)^3

basically what you said

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[gritty solar]

So for part (c) here, I thought of something but I need help making it rigorous

If you have a point x (irrational) in F, then it would be possible to find rationals a and b such that a < x < b, but due to the construction of O, there would be open intervals around a and b and these intervals would engulf the irrationals around x too, and thus x must be isolated?

drawing not to scale

[final saddle]

@gritty solar Has your question been resolved?

[gritty solar]

jan you have a dirty mind

[mint orbit]

sorry the image has a striking resemblance to my face i feel called out

[onyx peak]

Why'd x be isolated? What if the intervals contain x? What if there is some rational closer to x whose interval contains it?

[gritty solar]

I mean we choose x so that that doesn't happen

[onyx peak]

If it's possible for O to be perfect, then you cant always choose such x

[gritty solar]

If it was possible to be perfect then why would there be a second part to the question

I guess it would depend on the ennumeration chosen

[onyx peak]

indeed

[gritty solar]

But say for a specific ennumeration it doesn't work

[onyx peak]

i think that some enumerations will lead to non-perfect F and some to perfect

your goal is finding examples of both

[gritty solar]

I need to actually find ennumerations for Q then?

[onyx peak]

i think so

[gritty solar]

💀

But how would I go about even doing that

[onyx peak]

thats a good question

well I would try making just one isolated point

say sqrt2

any irrational will work

[gritty solar]

Okay

[onyx peak]

now the enumeration can be almost arbitrary, except for when the rationals are close to sqrt2. Youll have to carefuly work with those and make sure that you never get too close to sqrt2 with the intervals

basically the enumeration of rationals will contain more and more points closer and closer to sqrt2 the further you go

and somewhere inbetween, youll just add the remaining rationals

[gritty solar]

I'd think that if a rational is 1/2^n away from sqrt(2) then let it have an eps-nbhd of that size

But we can't do that from both sides

Wait lowkey if we just

Oh no

that wouldn't work

Because of the closure property of Q

😔

[onyx peak]

you techniically can

one rational from below, one rational from above, one random rational, one rational from below, one rational from above, one random rational...

[spring haven]

haha

[onyx peak]

as long as you can carefully enumerate the rationals from below, from above and all the remaining randoms (which is simple), then you can just use the enumeration above

[gritty solar]

Okay, I'll have to devise a system to do that

Okay, let's say we're looking at sqrt(2) and for the sake of this proof we only focus on [1, 2]

We can map 1 --> 2, and 2 --> 3

so the interval cuts to [1.25, 1.875]

now map 1.25 --> 4 and 1.875 --> 5

like that?

I feel like some axiom of choice bullshit might come up here

[onyx peak]

that could work, but you'd be leaving lots of rationals behind

which might not be that much of an issue actually

Q is countable

oh nvm it is an issue

the problem is that you'll then need to somehow enumerate all those rationals

[gritty solar]

who cares ennumerate them however

[onyx peak]

which you didnt enumerate in here. And the ones which are close to sqrt(2) cant be enumerated arbitrarily (arbitrary enumeration would make some of them have intervals overlapping with sqrt2)

[onyx peak]

and the ones close to sqrt2 will overlap with it

[gritty solar]

yeah

so like

we slowly back track

😭

[gritty solar]

thats why i said this

[onyx peak]

Yeah, the question is how exactly do we use choice

oh and we will also need to make sure that sqrt2 is actually isolated

i.e. that we have O everywhere around it

that's probably the easier part

[gritty solar]

"Pick an ennumeration such that sqrt(2) is isolated therefore it is isolated by construction"

[onyx peak]

Who cares if it exists lol

[gritty solar]

can I come back to this in about 20 minutes? I'll ping you

[onyx peak]

i gtg as well for 5 mins, ill try coming up with sth after that

[final saddle]

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[timber plume]

I confirmed my answer with differentiation short cut

they dont igve the same

answer

[last rampart]

are you sure that f is even

[final tangle]

missed that mistake earlier when looking over it

[timber plume]

oh did i mess q1?

[last rampart]

rmemebr that -(-t) = t

yea

[timber plume]

which one

[last rampart]

uhhh i haven’t finish reading so idk if it’s relevant to the next parts

[timber plume]

did i forget the bracket again

[timber plume]

all good, lmk when ur done

[last rampart]

sorry for the delay, but first principles is the definition of derivative right?

like the limit definition

im assuming you are confused on part b?

[final tangle]

sign issue mentioned in part a)
() issue in part b) and wrong expansion

[last rampart]

if you are confused on part b it is because you expanded 4(t+h)^2 wrong, should've gotten 8th as second term in numerator

my apologies

read the question wrong

[final saddle]

@timber plume Has your question been resolved?

[timber plume]

did i get al of them right this time

the one in red highlight is all part of Qd

wait 8t +1 is nto

[deep condor]

yep, but you wrote -1/17 under (c) instead of -1/15

[timber plume]

not equal to 8t-1

[deep condor]

oh i missed that ._.

[timber plume]

i noticed that rn haha

[deep condor]

definitely sub f(t+h) using brackets, you expanded -(t+h) = -t+h

[final tangle]

more () issues

also you should include taht h^2 term

[deep condor]

yeah i would write out the whole expansion

(a), (c), and (d) look good

[final tangle]

and f(t) isn't -f(-t)

[timber plume]

I cant spot it, do u mind if you could screenshot it

[final tangle]

$(t+h)^2 = t^2 + 2th \red{+h^2}$

[soft zealot]

ℝαμOmeganato5

[final tangle]

and for part a)
$$-f(t) = -(4t^2 - t + 2)$$
that doesn't then equal $-4t^2+t+2$ \
which isn't $f(t)$ either

[soft zealot]

ℝαμOmeganato5

[final tangle]

can't you do the work with pen/paper?

[timber plume]

I dont have

the print

[final tangle]

i suggest redoing the whole question from the beginning

wdym the print

[timber plume]

I can do it on paper but im tryna save some ink

[timber plume]

printed version

hard copy

[final tangle]

you can look at the question / copy it down
and do the calculations on paper

[timber plume]

yea for some reason they dont give us enough room to write

even in the real test

[final tangle]

how much do they actually give you? no separate writing booklet?

[timber plume]

It depends on the question, I think they can give writing booklet if we ask

I never asked

And also I finished Q5

Just want to confirm, there is only one stationary point on the curve correct?

[brittle breach]

You've got a doubt 24/7

Not complaining

[timber plume]

Lmao

[brittle breach]

If 4t² - t + 2, you can write it as 2t (t - 1) + 2

[timber plume]

No way, did I still make a mistake

[brittle breach]

2 🤷‍♂️

If t tends to 0, the function is equals 2

[timber plume]

Which q are you talking about

[brittle breach]

The below

Qb?

[timber plume]

I don’t understand

How does that relate

[brittle breach]

OK sorry

[timber plume]

to qb

[brittle breach]

I believe Qb is separate from Q5

[timber plume]

It’s part of it

[timber plume]

^

[brittle breach]

h in limits always tends to 0

Could even translate it to differentiation

d/dx

[final tangle]

looks ok now

[timber plume]

Thanks

.close

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[lofty canyon]

uhhh im not sure which topic specific channel is the most approriate for this q so imma jus ask it here- if we have a number h->0 then can we consider there are numbers between h and 0

[tired walrus]

well i suppose we can but that question sounds a bit weird

!xy

[final saddle]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[lofty canyon]

theres no specific problem question to show- it jus kinda entered my mind while i was studying limits

[lofty canyon]

alright but why cuz if its ->0 then isnt like elemental enough to be next to 0

[tired walrus]

"elemental"?

[lofty canyon]

small enough

[tired walrus]

???

[lofty canyon]

i meant small enough

[tired walrus]

i do not know what "small enough" is supposed to mean

[lofty canyon]

uhhhh its okaayay

im not sure how to explain it

[jagged flare]

isnt it a yes? consider h/2

[lofty canyon]

okay so if we have h->0 so then does tht mean h is really small or is it right next to 0

[magic coyote]

There is no "right next" to zero

There can always be other numbers inserted between h and 0

No matter how small h gets

That's how real numbers work

You might say 0.0000000001 is right next to 0

[jagged flare]

isnt there like a property

like R is dense or smth

[magic coyote]

Um

Idk but am just explaining in a logical way lol

[magic coyote]

But then 0.00000000001 is even closer

So you can never be right next to 0

[lofty canyon]

yea exactly

im saying

[jagged flare]

a subset E is dense in T iff for all a<b in T there exists x in E such that a<x<b of smth

thats what google says

[lofty canyon]

if h keeps getting closer and closer to 0 it means the number of numbers between h and 0 keeps reducing so then

okay no wait

ur right

thanks

[lofty canyon]

what does dense mean 😭

[jagged flare]

the second part of ny aentence after iff (atleast if i read it correctly)

[lofty canyon]

OHHH Okyayay got it

thankssss

.close

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[errant basin]

given the stereographic projection is a homeomorphism from R^n to the punctured ball in the higher dimension and the ith coordinate of the projection of (x1,...,xn+1) is x1/(1-xn+1), prove the ball is homeomorphic to the one-point compactification of the lower dimension space

[brittle breach]

Time to throw the punctured ball into the 7th dimension

[errant basin]

all my homies love the 178635289th dimension

anyways

i can send a screenshot of the answer and explain what parts i dont understand, but i will appreciate it very much if u could explain from the ground up

parts i dont understand:
in lines4 and 5, since S^n is a subset of E^n+1 ... implies 1-epsilon>xn+1

oh i only dont understand that part. neat

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@errant basin Has your question been resolved?

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[fringe sigil]

i was solving this and i got c is less than or equal to (-infinite,0] U [1,infinite)

is it possible to draw any conclusion from this??

[daring lion]

what do you mean by
less than or equal to (-∞,0)?

less than or equal to should be used with a number

like <= 4

did you mean that the expression is always real for c in (-∞, 0] U [1, ∞)?

[fringe sigil]

well i got an equation with, c is less than or equal to (quadratic)/(quadtratic)

and the (-infinite,0] thing was its range

[daring lion]

oh

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[errant basin]

parts i dont understand:
in lines4 and 5, since S^n is a subset of E^n+1 ... implies 1-epsilon>xn+1

[errant basin]

<@&286206848099549185>

[final saddle]

@errant basin Has your question been resolved?

[errant basin]

<@&286206848099549185>

i understand now, there is a small typo there.

.close

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[rancid depot]

This is the question I have to do and here is my force diagram, what exactly am I doing wrong? And is there an easier way to do it?

[wraith roost]

Bruh! Th C thing is not defined!

[lime crest]

torque? i dunno if thats what youre already doing

[final saddle]

@rancid depot Has your question been resolved?

[shut schooner]

yeah exactly

and if the length of the string and radius are the same, it would make a very very wierd diagram i think

[rancid depot]

Isn’t it still solvable?

[final saddle]

@rancid depot Has your question been resolved?

[brittle breach]

Radius of sphere = a
Length of string = a
Tension in the string = 2W/sqroot 3

[old quarry]

equate anticlockwise torque to clockwise torque

you can take about what ever point u want but i suggest take about C

[rancid depot]

okay ill try that thank you

[old quarry]

no wait sorry

take about A not C

cause u cant write torque for tension

[rancid depot]

okay ill do that thanks

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[cosmic sleet]

(243)^1/5

[tranquil pine]

,calc (243)^1/5

[soft zealot]

Result:

48.6

[cosmic sleet]

no

[tranquil pine]

ok

[terse crypt]

3^5 = 243

[terse crypt]

,calc (243)^(1/5)

[soft zealot]

Result:

3

[barren jetty]

Hi

[sullen gale]

hello, if you have a question check out #❓how-to-get-help . if you want to chat then look at #360643390594875392 or the discussion category in general. this channel is currently occupied by another user.

[barren jetty]

oh sry

[sullen gale]

np, welcome to the mathcord!

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[exotic shale]

hey is there any geometrical meaning of an integral over a curve of a 2 input function ?

[exotic shale]

like there is with reimann integral where we can interpret it as sum of rectangles with very small side dx

that would be surface integral

where you have dx dy

this you got ds on a curve

[tranquil pine]

oops didnt read that curve part

[vital crag]

Just show an example

[exotic shale]

is this what we refer to as contour integral?

[drowsy epoch]

Well in the context of complex analysis you would call it that, but this a real valued function so you'd just say line integral usually.

[exotic shale]

whats the difference

[drowsy epoch]

The contour lies in the complex plane

[exotic shale]

oh

can it be 3d?

[drowsy epoch]

What canbe 3D?

[exotic shale]

not 3d uhm i mean

if we use z for the imaginary axis

and x for real

can there also be a y axis

i mean like is there a case where curve is not on 1 plane but on a xyz coordinate system

[drowsy epoch]

There is this thing I know of

Like you can have real valued function, and add a third dimension to visualize the complex roots of it for instance of a quadratic, what is above only shows the relation between exp function and trigonometric functions.

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@exotic shale Has your question been resolved?

[exotic shale]

ok

yea alr thanks

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[round harbor]

Aye, how to find cosec 300000 degrees?

[atomic moon]

300000 mod 360 = ?

Take the result and put it in 1/sin(x)

Done

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[round harbor]

Thanks bro 😁

[atomic moon]

Yw

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[timber plume]

Hey, I’m not sure why I’m getting this wrong

[drowsy epoch]

What rule did you use?

[timber plume]

I use the power rule

[drowsy epoch]

But it seems you power ruled the wrong thing.

[old moon]

ik the power rule is fast but try using the definition yall

[drowsy epoch]

You would take down the 3/2 and subtract 1 from that power.

[worthy wren]

unless stated otherwise, it is always more convenient to use the rules we proved using the definition

[old moon]

ig so but math is about showing

[timber plume]

So I used it at the wrong time

[old moon]

like use the definiton for 3tan(sinx)

[magic coyote]

You dont sit and prove the power rule in every differentiation

[drowsy epoch]

I don't know, what time is it for you?

[old moon]

sometimes i do

[magic coyote]

weird

[old moon]

u gotta do wat u gotta do

[timber plume]

why am i keep forgetting simple algebra

[worthy wren]

uu, what were you trying to do here?

[old moon]

see thats why u use the def to upgrade ur algebra

[timber plume]

cuz if u differentiate x its 1

[drowsy epoch]

Broski, you said power rule, not product.

[worthy wren]

okay so if you have f(x)g(x) and you want to differentiate it, this is not equal to f'(x)g'(x)

[old moon]

i wish it was

[worthy wren]

if u havent learnt the product rule, use the method of universe

[old moon]

but no u use gf'+fg'

[worthy wren]

which is the power rule

[timber plume]

is that chain rule?

[worthy wren]

chain rule has to do with compositions of functions

[old moon]

chain rule is if there is a composition of functions

HA

almost beat ya

[timber plume]

makes sense

[worthy wren]

f(g(x)) is a rule that says we take g(x) then plug it into f(x)

we sometimes want to differentiate that

[drowsy epoch]

g(x) = x

[worthy wren]

that is when we invoke the chain rule

[old moon]

thats the indentity function pretty sure

[worthy wren]

yes that is the id map

[drowsy epoch]

Prove it 🫵🏻

[old moon]

wdym prove it, its already like that

[timber plume]

im confused what do u mean

[worthy wren]

for all x in D, if g(x) = x then thats the definition of identity

[drowsy epoch]

Why would you differentiate x seperately?

[timber plume]

why cant i differentiate x

[old moon]

number 1, u would be able to do it if it was a constant multiple

[drowsy epoch]

Because you did consider them as one thing x^3/2 not as a product

[old moon]

u kept the x

differentiated it

but u cant pick and choose

just like algebra

[timber plume]

true, okay i understand now

thanks

.close

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[timber plume]

.reopen

[final saddle]

✅

[old moon]

yes bot, slayyy

[timber plume]

what were you saying @tranquil pine

[timber plume]

real

[old moon]

💅

[tranquil pine]

I only saw the original qn 😂 so i wrote the exact same thing someone else said

f’(x) = fg’x + gf’x

[timber plume]

no worries

at all

.cllose

.close

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[arctic mica]

How do I approach with this without simplifying into sin and cos

[vital crag]

Did you try u=sqrt(tan(x))

[golden comet]

he did

[drowsy epoch]

Did I try it?

[vital crag]

Can you show what he did

[fiery bluff]

||you need sec^2 to cancel out tho?||

[vital crag]

No. Try u=log(x) too

[final saddle]

@arctic mica Has your question been resolved?

[fiery bluff]

Im imagining a u=theta-pi/4 sub

[arctic mica]

oo ur right

I was thinking of converting it into π/4 + θ by cancelling the minuses in num and denom but that didn't work

Lemme try that

That leaves a sec and tan in the numerator and a √tan in the denominator

Would it be wise to write the denominator as a form of sec and substitute that?

They're still different though

I don't see how a sub would work around it

[fiery bluff]

Combine that into the denom

Then you get something like $\sqrt{\sin 2\theta}$

[soft zealot]

Element118

[arctic mica]

Would it be possible to solve this integral without ever simplifying into sin or cos

[arctic mica]

cuz I wanted to solve this without simplifying into sin and cos terms

[fiery bluff]

eh why do you want to restrict yourself

[arctic mica]

Just looking for other methods

Already got the answer by deepseek in sin and cos terms but it seems to have trouble finding the correct answer without simplifying into sin and cos

[fiery bluff]

Well you can just work with tan theta/2 iirc

And that mirrors sin and cos

[arctic mica]

weierstrass sub?

[fiery bluff]

Yeah if you dont want to touch sin and cos you can always just tan theta/2 those expressions

[arctic mica]

Alright I'll try that approach

[ivory jacinth]

vibe mathematician

uses AI to do math

[arctic mica]

what

[ivory jacinth]

https://en.wikipedia.org/wiki/Vibe_coding

[arctic mica]

uh okay

[final saddle]

@arctic mica Has your question been resolved?

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[timber plume]

I'm trying to solve for x

but im a bit confused

am i atlaest on the right track?

[pliant shore]

careful: 1 - x^2 is not equal to -(1 + x^2) = -1 - x^2

[timber plume]

I see

how do i approach this q

[severe verge]

you can see that the denominator can never be 0

so the numerator must be 0

[timber plume]

so x is equal to1

[severe verge]

you missed a solution

[pliant shore]

1 - x^2 = 0 has two solutions

[timber plume]

got it

+1, -1

[pliant shore]

yep! that's correct

[timber plume]

my other question was with this particular problem

7. a

[severe verge]

product rule in derivatives

[pliant shore]

write out the product rule form where you leave the d/dx (...) for now

then d/dx (x^2) should be no problem

for d/dx (2x - 1)^3, you need the chain rule, which tells you to multiply the thing you get after the power rule by what?

or have you already found f'(x) but you haven't factored it yet

[final saddle]

@timber plume Has your question been resolved?

[final saddle]

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[iron panther]

can someone please explain this? why can we surely find such p, q and r?

[tired walrus]

consider the span of {1, sin(x), cos(x)} in the space of all functions from R to R

errrr wait hold on.

are we sure thats what we want the linear independence of

i think the linear independence of {d cos(x) + e sin(x) + f, -d sin(x) + e cos(x), 1} is rather more important

for which we need like... d and e both nonzero i think

which means they span the same space as {1, sin(x), cos(x)}

[iron panther]

what is meant by 'span in the space' ?

[tired walrus]

do you know what span means in linear algebra

[iron panther]

no

[tired walrus]

oh dear

... wait have you even done any linalg before

[onyx peak]

If you dont know much linalg, just skip it for now and jump to equating the coefficients. It will give you a linear system of equations and if you at least have some intuition about linear systems, you'll see that the system has a solution

[atomic moon]

If i say to you that R² = span((1,0), (0,1)) does that remind you something ?

[worthy wren]

if you do insist on knowing why we require linear independence, just know it forces unique solutions onto the coefficients

[atomic moon]

The coefficients being the coordinates in the basis of the span, so thats why its unique

(Two distincts vectors having same coordinates is not a thing)

[summer root]

What class is this?

[iron panther]

no it doesnt

[iron panther]

yeah i was thinking why cant it have more than one solutions

[atomic moon]

Span(.) is the space created by the linear combinations of what is in ()

So all the functions of the form acosx + bsinx + c for the span(1,cosx,sinx) and a,b,c being reals

[iron panther]

oh

[atomic moon]

Its like when you say that (5,3) in the canonical basis of R² is 5*(1,0) + 3(0,1) and so the coordinates are (5,3)

Same here but with functions instead of R² vectors

[iron panther]

and that function can never be equal to zero? so they are linearly independent?

for all values of x

[atomic moon]

Linearly independant means that you can't express one of the function with a linear combination of the others

And yeah for all values of x

[atomic moon]

More like that the function = 0 implies coefficient being all 0

For all x again

[iron panther]

yeah i had studied a bit of linear independence in vectors

[atomic moon]

Same concepts with functions then

[iron panther]

how can we prove it for sinx cosx and 1 ? it just feels like they are linearly independent but i dont exactly know why

[atomic moon]

Set it equal to 0

And choose wisely the value of x to get that all coefficients must be 0 if its equal to 0

You can look at limits, derivative also but here it doesn't brings much since sin and cos do not have limits and the derivatives are cycle

[onyx peak]

the derivatives actually can help here i think

but there is like a thousand ways to prove it

[atomic moon]

Ah to get rid of the constant

True

[iron panther]

i can make a degree 2 equation in sin x and if both roots of the quadratic are either less than -1 or more than 1 then maybe all the coefficients will have to be equal to zero

[atomic moon]

"Maybe" no, be sure

Can take the derivative and see what happen in x = pi/2 for example or x = 0

[iron panther]

-asinx +bcosx = 0
tanx=b/a
we found a solution other than a,b,c = 0 so its linearly independent?

[atomic moon]

No

Linearly independant means acosx + bsinx + c = 0 ==> a,b,c = 0

For all x in R

tanx not being defined on all x in R this do not work

So -asinx + bcosx = 0, if we take x = 0 we have :
-asin(0) + bcos(0) = 0
bcos(0) = 0
b = 0

Ok

So now we can fast deduce that -asin(x) = 0 and so a = 0 by going back to the derivative

[iron panther]

yeah

[atomic moon]

So now the initial equation is acos(x) + bsin(x) + c = 0 but we have seen that a = b = 0 with the derivative

[iron panther]

so c = 0

[atomic moon]

So c = 0 comes neccessarly

And we have proved linearly independance

[iron panther]

so because s,c,1 are linearly independent
we get
a = pd + qe
b = pe -qd
c = pf + r

from the first eqn we get one value of p,q

and from the third one value of r

so theres only 1 solution?

[atomic moon]

independant but yeah

If its dependant there is infinitely many sol

And you can't make the equations you wrote

[iron panther]

typo

[iron panther]

wdym?

[atomic moon]

(If it was dependant)

What you did is good

[iron panther]

yeah

thanks

[atomic moon]

Yw

[iron panther]

also, i had once learned how to find if a system of three equation has no solutions/1 solutions/inf solutions using determinants
i know the method but not the proof/ logic of why it works
do you know where can i learn it properly

online

i think its related to this

[final saddle]

@iron panther Has your question been resolved?

[final saddle]

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[brazen stump]

the correct answer is the black bit my answer is the green bit why would my answer be wrong and for the black ink working out why is it correct why do we just multiply?

[severe verge]

you completely misunderstood the problem

[raven marsh]

70% OF the 80% wanted big screen

[storm haven]

draw a diagram

venn diagram maybe

[raven marsh]

what diagram is there

huh

[storm haven]

venn

[raven marsh]

thetes no need to overcomplicate

its an easy principle

[storm haven]

helps visualize for begginers no?

[brazen stump]

yh so wouldnt it be 0.7/0.8

[raven marsh]

okay so

[brazen stump]

since is OF

[raven marsh]

70% said yes

0.70 said yes

[brazen stump]

yh

[raven marsh]

You okay with this?

okay

so 80% OF the people who said yes

which is 80% * 0.7

Want a big screen

[brazen stump]

oh yh

[raven marsh]

and can you calculate that

[brazen stump]

so 0.8 x 0.7